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Directed paths decomposition of complete multidigraph. Zdzisław Skupień Mariusz Meszka AGH UST Kraków, Poland. For a given graph G of order n, th e symbol λ G stands for a λ-multigraph on n vertices, obtained by replacing each edge of G by λ edges (with the same endvertices). - PowerPoint PPT Presentation
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Directed pathsdecomposition of
complete multidigraph
Zdzisław SkupieńMariusz Meszka
AGH UST Kraków, Poland
For a given graph G of order n, the symbol λG stands for
a λ-multigraph on n vertices, obtained by replacing each edge
of G by λ edges (with the same endvertices).
G 4G
If G Kn then the symbol λKn denotes the complete λ-multigraph
on n vertices.
A decomposition of a multigraph G is a family of edge-disjoint
submultigraphs of Gwhich include all edges of G.
Theorem [M. Tarsi; 1983]Necessary and sufficient conditionsfor the existence of a decomposition of λKn into paths of length m areλn(n-1) 0 (mod 2m) and n m+1.
[C. Huang][S. Hung, N. Mendelsohn; 1977]handcuffed designs
[P. Hell, A. Rosa; 1972]resolvable handcuffed designs
Theorem [M. Tarsi; 1983] The complete multigraph λKn is decomposable into undirected paths of any lengths provided that the lengths sum up to λn(n-1)/2, each length is at most n-3 and, moreover, n is odd or λ is even.
[K. Ng; 1985] improvement on any nonhamiltonian paths in the case n is odd and λ=1
n=9λ=1
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0 1 7 2 6 3 5 4 1 2 0 3 7 4 6 5 2 3 1 4 0 5 7 6 3 4 2 5 1 6 0 7
Conjecture [M. Tarsi; 1983] The complete multigraph λKn is decomposable into undirected paths of arbitrarily prescribed lengths provided that the lengths sum up to λn(n-1)/2.
For a multigraph G, let DG denote a multidigraph obtained from G
by replacing each edge with two opposite arcs connecting
endvertices of the edge.
G DG
For a given graph G of order n, the symbol λG stands for a λ-multigraph
on n vertices, obtained by replacing each edge of G by λ edges (with the same endvertices).
The symbol λDKn denotes the complete λ-multidigraph on n vertices.
digraph D
λ-multidigraphon n vertices, obtained by replacing
each edge of G by λ edges (with the same endvertices).(with the same endvertices).
arc of D arcs
λ-multidigraph
λD
G 4G
DG 4DG
A decomposition of a multigraph G is a family of
edge-disjoint submultigraphs of Gwhich include all edges of G.
A decomposition of a multigraph Gmultidigraph D
which include all edges of G.arc-disjoint submultidigraphs of D
arcs of D
Problem [E. Strauss; ~1960]Can the complete digraph on n vertices be decomposed into n directed hamiltonian paths?
[J-C. Bermond, V. Faber; 1976]even n
[T. Tillson; 1980] odd n, n 7
Theorem [J. Bosák; 1986] The multigraph λDKn is decomposable into directed hamiltonian paths if and only if neither n=3 and λ is odd nor n=5 and λ=1.
Problem [Z. Skupień, M. Meszka; 1997] If the complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths then the lengths must sum up to λn(n-1), and moreover all paths cannot be hamiltonian if either n=3 and λ is odd or n=5 and λ=1.Are the above necessary conditions alsosufficient for the existence of a decomposition into given paths?
Theorem [Z. Skupień, M. Meszka; 1999]For n 3, the complete multidigraph λDKn is decomposable into directed nonhamiltonian paths of arbitrarily prescribed lengths ( n-2) provided that the lengths sum up to λn(n-1).
Theorem [Z. Skupień, M. Meszka; 2004]For n 4, the complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths except the length n-2, provided that the lengths sum up to λn(n-1), unless all paths are hamiltonian and either n=3 and λ is odd or n=5 and λ=1.
Corollary [Z. Skupień, M. Meszka; 2004] Necessary and sufficient conditions for the existence of a decomposition of λDKn into directed paths of the same length mare λn(n-1) 0 (mod m) and m n-1,unless m=n-1 and either n=3 and λ is odd or n=5 and λ=1.
Conjecture [Z. Skupień, M. Meszka; 2000] The complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths provided that the lengths sum up to λn(n-1), unless all paths are hamiltonian andeither n=3 and λ is odd or n=5 and λ=1.
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